On the Complexity of Computing Zero-Error and Holevo Capacity of Quantum Channels

Salman Beigi; Peter W. Shor

Home/Archive/arXiv

arXiv · 0709.2090

On the Complexity of Computing Zero-Error and Holevo Capacity of Quantum Channels

Salman Beigi·Peter W. Shor

View original source ↗Read source document ↗Export metadata ↓

Abstract

One of the main problems in quantum complexity theory is that our understanding of the theory of QMA-completeness is not as rich as its classical analogue, the NP- completeness. In this paper we consider the clique problem in graphs, which is NP- complete, and try to find its quantum analogue. We show that, quantum clique problem can be defined as follows; Given a quantum channel, decide whether there are k states that are distinguishable, with no error, after passing through channel. This definition comes from reconsidering the clique problem in terms of the zero-error capacity of graphs, and then redefining it in quantum information theory. We prove that, quantum clique problem is QMA-complete. In the second part of paper, we consider the same problem for the Holevo capacity. We prove that computing the Holevo capacity as well as the minimum entropy of a quantum channel is NP-complete. Also, we show these results hold even if the set of quantum channels is restricted to entanglement breaking ones.

Explore related subjects

quant-ph

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Salman Beigi, Peter W. Shor. 2008-10-13. On the Complexity of Computing Zero-Error and Holevo Capacity of Quantum Channels. https://arxiv.org/abs/0709.2090

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Entanglement Detection with Rotationally Covariant Measurements -- From Compton Scattering to Lemonade

The accurate and efficient detection of quantum entanglement remains a central challenge in quantum information science. In this work, we study the detection of entanglement of polarized photons for measurement devices that are specified by rotational covariance. We derive explicit positive operator-valued measures (POVMs) showing that any such (idealized) setting is classified by one real measurable parameter r. In particular, we give a POVM formulation of the Klein-Nishina formula for Compton scattering of polarized photons. We provide an SDP-based entanglement certification method with computable bounds that operates on the full measured statistics, also considering semi-device independent scenarios. Furthermore, we show that, while Bell violations are impossible with two rotationally covariant measurements, EPR steering can still be certified under one-sided symmetry constraints. Finally, we demonstrate how our approach can be applied by analyzing a rotationally covariant showcase experiment using scattered polarized light in a selection of soft drinks. Our results motivate further investigation of entanglement detection with lemonade- based detectors.

quant-ph↗

arXiv2026

Simulation of topological superconductors and their competing orders using photon-mediated interactions

Realizing and controlling the unconventional pairing featured by topological superconductors remains a central challenge. We introduce a cavity QED quantum simulator that engineers competing chiral $p_x+ip_y$ and $d_{x^2-y^2}+id_{xy}$ orders by tailoring cavity-mediated couplings between atomic pseudospins that emulate momentum-dependent pairing channels. The desired spatially inhomogeneous cavity-mediated couplings can be engineered in a 2D optical lattice using incommensurate cavity-lattice wavelengths naturally occurring in cavity QED systems. This minimal and fully tunable platform enables controlled state preparation and continuous measurement of superconducting order parameters. To probe this competition, the platform enables selective excitation and direct time-domain detection of Bardasis-Schrieffer modes---collective excitations of the subdominant pairing channel. The same competing channel can have an even stronger impact out of equilibrium, where a sudden quench can drive its growth and abruptly reshape the dominant superconducting order, despite the stability of the dominant order in equilibrium. Together, these results establish cavity QED as a controlled setting for revealing hidden competing orders and exploring nonlinear dynamics of topological superconductivity that remain elusive in solid-state and ultracold-atom systems.

quant-ph↗

arXiv2026

Contractivity of time-dependent driven-dissipative systems

Experimentally relevant quantum systems are frequently described by a time-dependent Lindblad master equation in which the dissipation due to interaction with an environment is fixed, but the experimenter has control over the Hamiltonian, which therefore can change with time. A fundamental question that is also of practical relevance is to determine when such systems invariably lose information about their initial state (are exponentially contractive independent of the Hamiltonian) or when the control can be used to prevent this loss of information to some degree. Here, we identify the necessary and sufficient criterion on the dissipator for Hamiltonian-invariant contractivity. If the dissipator does not fulfil this criterion, but by itself is exponentially contractive, we also establish exponential contractivity for sufficiently weak or slow drives and construct examples where strong or fast driving prevents exponential contraction.

quant-ph↗